《Fundamentals of Computer Graphics》

Lecture 7 OpenGL 3D Transformation

and Fractal Geometry

Yongjin Liu

What about this lecture？

Scene layout of OpenGL

Three transformation matrix

glMatrixMode

Viewport and window of OpenGL

viewport

window

Contents

Transformation and observation control in

OpenGL

Provide a set of powerful and flexible functions

Linear algebra

Difference between viewport transformation

and projection transformation

Coordinate and matrix

Matrix operation command

Transformation pipeline

Geometry transformation pipeline

Object Coordinate

Model Transformation

World Coordinate

Projection Variable

Perspective division

ViewPort Transformation

Window Coordinate

ModelView Matrix

Clipping

Coordinate Normalization

View Transformation

Pixel

data

Vertex

data

OpenGL 3D transformation

Scene layout of OpenGL

Three transformation matrix

glMatrixMode

Viewport and window of OpenGL

viewport

window

Up direction

Observe direction

Near plane

Far plane

World coordinate

Viewing volume

Matrix transformation of OpenGL

Model transformation

View transformation

Projection transformation

Near plane

Far plane

World coordinate

Viewing volume

Matrix transformation of OpenGL

Model Transformation

View Transformation

Projection Transformation

Matrix transformation of OpenGL

Model Transformation

View Transformation

Projection Transformation

glMatrixMode(parameter)

glMatrixMode(GL_MODELVIEW)

glMatrixMode(GL_PROJECTION)

Transformation from world window to viewport

screen

World window

viewport

screen

World window

viewport

glViewport(x, y, width, height)

Transformation from world window to viewport

Example analysis

1、Initialize window

glutInit(&argc, (char**) argv);

glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);

glutInitWindowSize(640,480);

glutInitWindowPosition(100,150);

glutCreateWindow("Sinc Function");

2、drawing function

void myDisplay(void) {// draw sinc function using world coordinate glClear( GL_COLOR_BUFFER_BIT );

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glBegin(GL_LINE_STRIP);

for(GLfloat x = -4.0; x < 4.0; x += 0.1)

glVertex2f(x, sin(pi * x) / (pi * x));

glEnd();

glFlush();

}

3、Initialize status of drawing

myInit(void) {

glClearColor(1.0,1.0,1.0,0.0);

glColor3f(0.0f,0.0f,1.0f);

glLineWidth(1.0);

}

4、Define world window

void setWindow(left, right, bottom, top) {

glMatrixMode(GL_PROJECTION);

glLoadIdentity();

gluOrtho2D(left, right, bottom, top);

//defines a 2D orthographic projection matrix

}

screen

World window

viewport

glViewport(x, y, width, height)

Transformation from world window to viewport

screen

World window Multiple viewport

Application window

Example analysis

Drawing hexagonal vortex

void render() {

glClear(GL_COLOR_BUFFER_BIT);

hexSwirl();

glutSwapBuffers();

glFlush();

}

Drawing function

void myDisplay(void) {

//set world window

glMatrixMode(GL_PROJECTION);

glLoadIdentity();

gluOrtho2D(140, 460, 40, 360);

int i, j, L=160;

for (i=0; i<4; i++) {

for (j=0; j<3; j++) {

glViewport(i*L, j*L, L, L);//set viewport } } glutSwapBuffers(); }

(140,40)

(460,360)

Width 320

Height 320

screen

World window Multiple viewport

Application window

Topic 1

Recursive method and

fractal geometry

Fractal geometry

Create object using simple program

• Using the self-similarity of real world object

• Using the mathematical recursive method

Fractal was first proposed by Mandelbrot

• Create a branch of math

• Can generate and simulate many interesting objects

• Can not be achieved using general geometry

modeling method

A simple example

Iteration rule

Koch curve and Koch snow

fractal geometry

Two feature measure

• Dependence of the unit of measurement for geometry object

• Self-similarity for geometry object

Scale and length

• Drawing resolution

• Minimal unit used in measurement

• The level of detail depends on the distance between us and the

object

Variable-length principle of Koch curve

• Segment of length 1 can be substituted by 4 off lines of

length 1/3

• For every iteration, the length of Koch curve increases 4/3

• Consider the limiting case (infinite iteration)

• Infinite curve, the first order derivative of each point is not

continuous

• It is not the usual one-dimensional curve, and not two-

dimensional object

• Fractal dimensional

Definition of fractal dimension

• Consider segment, square, cube of unit length

• The resolution of iteration h = 1/n, n is integer

• Splice the segment into k = n parts

• Splice the square into k = n2 parts

• Splice the cube into k = n3 parts

Definition of fractal dimension

• From the mathematical point of view, dimension is used

to characterize the geometric properties of the object

• To any object（segment, square, cube), there exists

equation：k = nd

• Solve k from the above equation, define d to be fractal

dimension: ln

ln

kd

n

Fractal dimension of Koch snow

• In the limiting case, the curve has infinite length, the

first derivative of each point is not continuous

• It is neither the general one dimension curve, nor the

2D object

• Segment with length 1 can be substituted with 4 off line

with 1/3 length

• n = 3, k = 4

• Fractal dimension

ln ln 41.26186

ln ln3

kd

n

Hilbert fractal curve

• Mathematician Hilbert discovered a very strange curve

• Can be proved that：

• The same as Koch curve，Hilbert curve extends

infinitely

• Hilbert curve never self-cross，but is limited in a box

• In the limiting case, Hilbert curve fill every point in the

box

• This curve is called space-filling curve

Hilbert fractal curve

• Iteration rule

Zero Hilbert curve

first Hilbert curve

first Hilbert curve

first Hilbert curve

second Hilbert curve

third Hilbert curve

third Hilbert curve

fourth Hilbert curve

David Hilbert (1862～1943)

• One of the most great mathematicians in the end of 19th

century, early of 20th century

• Fractal method，Dirichlet principal，methods of

mathematical physics

• 23 mathematical problem of Hilbert

– The second international congress of mathematicians in Pairs,

1900

– The eighth problem, prime problem，Jingrun Chen

– May 24, 2000，College de France in Pairs，announced 7 new

millennium mathematics problem, 1 million dollar bonus for each

– Poincare guess，P and NP problem, etc.

Fractal geometry

Use simple basic shape

Use simple iteration and refinement

process

Obtain complicated nature object

Uniform mathematical description

of iterative process

Iterated function system (IFS)

• d0

• d1 = f(d0)

• d2 = f(d1) = f(f(d0))

• d3 = f(f(f(d0)))

• … …

• dk = f [k](d0)

d0, d1, d2, d3, …is orbit of d0

Iterated function system (IFS)

• f (x) = 2x

• f (x) = cos(x)

• f (x) = 4x(1x)

• f (x) = x2 + x

Uniform mathematical description

of iterative process

Golden ratio and IFS

Ancient Greek Parthenon

Golden ratio

11

1 51.618033989

2

Golden ratio and IFS

1 1 1 1 ( ) 1

1 11 ( )

1 11

11

1

f x x

f xx

Golden rectangle

Golden triangle

Golden Rhombus

Golden Rhombohedron

Golden rectangle

Logarithmic spiral baer

Golden triangle a

b

Golden diamond

Golden rhombohedral

How to draw fractal geometry

shape effectively?

Turtle Graphics

turn(float angle)

forward(float dist, bool isVisible)

Example analysis polyspiral

For (some iterations) { forward(length, 1); turn(angle); length += increment; }

for(int i=1; i<3; i++) { forward(L, 1); turn(60); forward(L, 1); turn(120); forward(L, 1); turn(60); forward(L, 1); turn(120); }

L-System and instruction generation

‘F’ means forward(1, 1)

‘+’ means turn(A)

‘’ means turn(A) F FF++FF

A simple example

Iteration rule

Koch curve and Koch snow

+, no substitution

atom：b

Generative form：b->a, a->ab

In mathematics, logic and

computer science, formal

language is a precise

mathematics or machine

that can handle language

defined by formula.

The same as languages in linguistics，formal

language generally has two aspect: grammar and

semantic.

The branch of mathematics and computer

science specializing in the syntax is called

formal language theory, it only committed to

syntax of language but not semantic.

In formal language theory, formal language is

a set of strings composed by finite letter.

A formal language can contain infinite number

of strings.

Formal definition of language

Operation between languages

A formal language can limit itself by many means:

Enumerate every string（applies only to finite string set）

Generate through formal grammar（see Chomsky pedigree）

Generate through regular expression

Recognize through certain automatic machine，such as

Turing machine, finite state machine

Representation method of language

Formal grammar, formal method, formal science,

formal system

Mathematical symbol, programming language

Extension of instruction set

Add other letter to instruction set, such as

X, Y

Only play a role in the iterative generation

process

Has no effect to Turtle motion

Dragon curve

‘F’‘F’

‘X’‘X+YF+’

‘Y’‘FXY’

Original instruction ‘FX’

‘FX+YF+’

‘FX+YF++FXYF+’

Extension mode of instruction set

(original instruction, F extension mode, X extension

mode, Y extension mode, rotation angle)

Add branch into instruction set

And instruction ‘[’ and ‘]’ into instruciton set

‘[’: saveTurtle() records the current position of

turtle

‘]’: restoreTurtle() restores the position of turtle

Implement using stack

‘[’: push the position of turtle

‘]’: pop the position of turtle

Example analysis

Original instruction ‘F’

Rotation angle 22o

Extension mode of instruction set

‘F’ ‘FF-[-F+F+F]+[+F-F-F]’

Bush

bush after 4 iteration

Random and growing refinement of

object shape

Instruction set with branch is still too regular

Introduce a small random number at every

‘+’, ‘’

Set different line width according the depth

of stack

Inheritance, mutation, crossover

Application of

Stochastic L system

L system in 3D space

Control the spatial orientation of turtle

Example design of 3D bamboo (Botany)

Bamboo generally has 4～8 branches

New branch grows on every branch

New branch generally has 4 growing direction

The angle between two branches is approximately

90o

The angle between branch and trunk is

approximately 33.4o

(9033.4)/90 = 0.618

Example design of other 3D plants

Design the number of branch and trunk

Design the growing direction of new branch

Design the angle of two branches

Design the angle of branch and trunk

Design the affine transformation matrix

Design of 3D herb

Design of 3D poplar

Design of 3D willow

Other example

maple

pine

camphor

Extension of theory system

Biology organism evolution:

lower, middle, higher levels

Basic L-System

Extended L-system (X/Y instruction,branch [])

Stochastic L-System

3D L-System

Content-based L-System

Content-independent L-System（3D）

Content-based L-System

2L-System

(k,l)-system

Extension of theory system

Basic L-System

Introduce branch

Stochastic L-System

3D L-System

Content-based L-System

Parameterized L-System

Parameterized L-System

Parameterized 2L-System

Application of L-System

Modeling of Nature

• D Cohen. Computer simulation of biological pattern generation processes. Nature, 216: 246-248, 1969

• H Honda, J.B.Fisher. Tree branch angle: maximizing effective leaf area. Science, 199:888-890, 1978

Application of L-System

Modeling of tree

Modeling of herb

Application of L-System

Modeling of tree

Modeling of herb

Leaf arrangement

Defining the pattern

of different species

More flowers and fruitages

Tsinghua historic site: moonlight over the lotus pond

More fruitage: pineapple, pinenut, etc

Flower bud

Application of L-System

Modeling of tree

Modeling of herb

Leaf arrangement

Automatic modeling of architecture

L system: Inheritance, mutation, crossover

Application of L-System

Modeling of tree

Modeling of herb

Leaf arrangement

Automatic modeling of architecture

Modeling of city

L system: Inheritance, mutation, crossover

The Palace Museum

Bibliography

Michael Barnsley.

Fractals everywhere, O177.3 FB26

Fractals in multimedia, O189.12 FF79

Superfractals, TP391.41 FB26

Spiral fern leaf

Fractal landscape

Fractal Planet

Life-form in abyssal region

Ancient

Art of mathematics？

Mathematics of art？

Week 13: this week

Week 14: one lecture

Week 15: everyone gives a presentation for your

course projects

Week 16: final mark evaluation